Questions: The Yoneda Lemma

Question 1

If F: C → Set is any functor and A is an object in C, the Yoneda lemma gives a bijection between Nat(Hom(A,-), F) and which set?

• A. The set of all functors from C to Set
• B. The set F(A)
• C. The set of morphisms from A to itself
• D. The set of all objects in C

Answer: (B)

The Yoneda lemma states Nat(Hom(A,-), F) ≅ F(A). Any natural transformation η: Hom(A,-) ⇒ F is uniquely determined by the image of id_A under the component η_A: Hom(A,A) → F(A). This single element of F(A) encodes the entire natural transformation — the rest is forced by naturality.

Question 2

The Yoneda lemma's bijection Nat(Hom(A,-), F) ≅ F(A) primarily holds when F is itself a representable functor.

• T. True
• F. False

Answer: False

The bijection holds for *any* functor F: C → Set — representability of F is not required. If F happens to be representable (F ≅ Hom(B,-) for some object B), that gives additional structure, but the Yoneda lemma applies universally to all functors into Set. This generality is precisely what makes it so powerful.

Question 3

What does it mean for the Yoneda embedding to be fully faithful, and why is this philosophically significant for category theory?

Model answer: A functor is fully faithful when it is bijective on hom-sets. The Yoneda embedding being fully faithful means morphisms A → B in C correspond exactly to natural transformations Hom(A,-) ⇒ Hom(B,-). Philosophically, this encodes the principle that an object is completely determined by its relationships to all other objects — you lose no information about C by studying it through its representable functors.

Full faithfulness is the precise statement that C embeds into the presheaf category without loss of information. Two objects that look the same from the outside (same representable functor) must be isomorphic inside C. This 'objects are determined by their morphisms' principle is one of the deepest ideas in category theory and motivates much of topos theory and sheaf theory.